Why are all inverse functions a One-To-One Function??

Question

seratul · Answer

One-To-One functions are functions that are true for both ways.
For instance- All x values have only one y value, and all y values have only one x value.

tkhunny · Answer

$f(x)$ must pass the vertical line test.
$f^{-1}(x)$ must pass the horizontal line test.
$f(f^{-1}(x)) = x$
$f^{-1}(f(x)) = x$
There's an explanation in there, somewhere.

seratul · Answer

Not sure what that explanation is xD.

seratul · Answer

In my notes, I wrote that A function only has an inverse if it is 1 to 1. Otherwise, it is just a reflection of y=x. Do you know what that means?

seratul · Answer

I should've asked but I thought I understood it. But as I'm doing the homework, I can't seem to understand the logic behind that.

tkhunny · Answer

Lean more on the word "function".

seratul · Answer

That doesn't really help but okay :'(

mathmale · Answer

In this situation, "one to one" characterizes a function:  For every input (within the domain) to a function, there is exactly ONE output.  Such a function has an inverse.

On the other hand, if for some particular input to the relation (not function), there are two or more output (y-) values, the relation is NOT one to one and does not have an inverse.

mathmale · Answer

If you start with a function f(x) and find its inverse, the inverse has to be one to one.  Else, you could not work backward from the inverse function to obtain the original function.

seratul · Answer

Okay, I think I got it, thank you.